Budapest was the country of birth of Franz Liszt and Bela Bartok the famous musician and Erno Rubik (who was also a sculptor and professor of architecture) the creator of Rubik Cube.
The best part about Budapest is that the tram are available 24 hours.
Some of the top sights listed in the map from information centre.
Chain Bridge - The most famous among the bridges connecting Buda and Pest
Parliament - Grand, massive, stunning
St Stephen's Basilica - "The mummified right hand of Hungary first king, St Stephen is kept in the chapel". When I was there I was sent away for restoration. From the picture, I saw that the hand is not in good condition as if its in the process decomposing.
Andrássy Avenue - Budapest answer to Avenue des Champs-Élysées
Dohany Street synagogue
Margaret Island - You can walk to the island from one of the bridge (Margit Hid). Very lively at night as there are clubs that open till wee hour of the morning.
Szenchenyi Spa and swimming pool - the experience was definitely worth it.
Gellert Hill and the Citadel
Museum of Fine Arts
Museum of Applied Arts
Palace of Arts
Museum of Ethnography
Memento Park - Statue Park
National Opera House
Other things that I would personally recommend:
Market (near Szabadsag Hid)
On the 2nd floor, there's a stall selling Hungarian cuisine. You have to try at least something, goulash, stuffed cabbage, paprika.
When you are done eating amuse yourself by trying to open the Secret Box. I had so much fun learning to open the boxes. Boxes come in a few sizes.
The river cruise (this is definitely better than the Seine river cruise in Paris)
What can I say, you get to see of the bridges and landmarks on either side of Buda and Pest up close. The parliament and Buda Castle really stood out. The cruise comes with 2 free drinks and you can also opt for dinner with additional price.
Some interesting info
Hid means bridge
Kirai means Chinese so you know what is a Kirai shop now.
I think the people in Budapest like to eat salty food.
For more on Hungarian cuisine.